## How to validate the correct software implementation of AHP?

Sometimes I receive the question, how to validate the correct implementation of my AHP excel template or my AHP online software, i.e. are the results correct and reliable? There are a number of steps to test the correct implementation of the method:

1. Check for a 2 x 2 decision matrix.
2. Use “1” for all elements of the decision matrix, i.e. each criterion has the same importance as all other criteria.
3. Use extremes, i.e. one criterion is extremely (9 times) more important than all other criteria.
4. Take specific examples from the literature – published research papers containing data of the decision matrix and priorities – and compare.
5. Compare the results for same input data with the results from another software implementation of the AHP method.

This is what I have done, and in the following I will explain each step in more detail.

## 1. Check for a 2 x 2 decision matrix

For two criteria there is only one comparison and one solution only: If criterion A is x-times more important than B, the weight w(A) = x/(1+x). w(B) = 1-w(A) as w(A)+w(B) = 1. Eigenvalue = 2; CR is always 0.

Example: Criterion A is 3 times more important than criterion B:
w(A) = 3/4 (75%), w(B) = 1/4 (25%). Check: w(A)/w(B) = 3.

## 2. Each criterion has the same importance as all other criteria

If all criteria have the same importance, the resulting weight should be 1/n, with n the number of criteria.

Example: Four criteria should result in a weight of 1/4 = 25% for each criterion.

## 3. One criterion is 9 times more important than all other criteria

If one criterion is 9 times more important than all other criteria, the weights depend on the number of criteria, the maximum weight or maximum priority wmax is always

wmax = M/(n + M – 1)

with M = 9, the maximum of the AHP scale, and n the number of criteria. All other weights should be

wmin = (1 – wmax )/(– 1)

Example: 5 criteria, one criterion 9 times more important than all others

wmax = 9/(5 + 9 – 1) = 9/13 = 69.2%

wmin = (1 – 69.2% )/(5 – 1) = 0.31/4 = 7.7%

## 4. Specific examples from the literature

Here a practical example comparing the results with  an example (7 criteria) given by Saaty in Int. J. Services Sciences, Vol. 1, No. 1, 2008 (p 86, table 2). The AHP matrix is:

 1 9 5 2 1 1 1/2 1/9 1 1/3 1/9 1/9 1/9 1/9 1/5 3 1 1/3 1/4 1/3 1/9 1/2 9 3 1 1/2 1 1/3 1 9 4 2 1 2 1/2 1 9 3 1 1/2 1 1/3 2 9 9 3 2 3 1

The result according Saaty is
(0.177,  0.019, 0.042, 0.116, 0.190, 0.129, 0.327) with consistency ratio of 0,022

My AHP Excel template and my online software should give the same results.

## 5.  Comparison with other software implementations

As I have implemented AHP under Excel and written in php script language for the online version, I can simply compare the results from both implementations, using the same input data.

## Welcome to Business Performance Management – Aug 2015

### Concepts, Methods and Tools to manage Business Performance

Dear Friends, dear Visitors,

time for another update of my welcome page! Since Chinese New Year in February there have been only two minor updates of my AHP Excel template. On request of many users, it can now handle two criteria. Though there are still a few requests to increase the number of criteria to more than ten, I am not planning to extend the template, as too many pairwise comparisons have to be done, which will be troublesome for the decision makers. It is much better to further breakdown the hierarchy into additional categories.

My AHP online  software seems to be quite stable. Beside the regular user administration, I didn’t put in  more effort, and the number of active users varies between three and four hundred.

I still get regular feedback via the contact page. It seems that many of my visitors are students, using my tools in their thesis. Sometimes I am asked, whether it is allowed to use them, and the answer is “yes”. As long as you use the software and templates for non-commercial purposes, e.g. education or research, you are allowed to do so. But please give credit to the author, and make a reference to the source. Under BPMSG-references you find a list of publications using my work.

Since end of last year I started a new hobby, flying remote controlled multi-copter. The header picture of my blog shows the skyline of Singapore, taken with this type of aerial platform. It is a fascinating technology and a lot of fun. I managed to finish my first own-build project, as small miniquad, and also started to develop some simple formulas to estimate flight speed and rate of climb. More to come here!

For now, please enjoy your visit on the site and feel free to give me comment – it is always appreciated.

Klaus D. Goepel,

Singapore, August 2015

BPMSG stands for Business Performance Management Singapore. As of now, it is a non-commercial website, and information is shared for educational purposes. Please see licensing conditions and terms of use.